Hereditary G-compactness

We introduce the notion of hereditary G-compactness (with respect to interpretation). provide a sufficient condition for poset not be hereditarily G-compact, which we use show that any linear order is G-compact. Assuming long-standing conjecture about unstable NIP theories holds, this implies an theory G-compact if and only it stable (and by result Simon, holds unconditionally $$\aleph _0$$ -categorical theories). G definable over A in theory, then $$G^{00}_A=G^{000}_A$$ . also include brief survey conditions G-compactness, with particular focus on those can used prove or disprove some (classes of) theories.